Stability and nonlinear wavy regimes in downward film flows on a corrugated surface

The linear and nonlinear stability of downward viscous film flows on a corrugated surface to freesurface perturbations is analyzed theoretically. The study is performed with the use of an integral approach in ranges of parameters where the calculated results and the corresponding solutions of Navier-Stokes equations (downward wavy flow on a smooth wall and waveless flow along a corrugated surface) are in good agreement. It is demonstrated that, for moderate Reynolds numbers, there is a range of corrugation parameters (amplitude and period) where all linear perturbations of the free surface decay. For high Reynolds numbers, the waveless downward flow is unstable. Various nonlinear wavy regimes induced by varying the corrugation amplitude are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 110–120, January–February, 2007.     

Quasi-geostrophic motions in a rotating layer of an electrically conducting fluid

Large-scale nonlinear oscillations of an electrically conducting ideal fluid of varying depth are considered with the magnetic, Archimedean, and Coriolis forces taken into account. The main equations are derived from an analysis of the scales of quasi-geostrophic motions. Under the assumptions that the Rossby numbers (a measure of the ratio of the local and advective accelerations to the Coriolis acceleration) are of the same order, the problem is reduced to a system of three nonlinear equations for hydromagnetic pressure and two functions describing the magnetic field. For an infinitely long horizontal layer of an electrically conducting rotating fluid, the exact solution of the corresponding nonlinear equations and the dispersion relation are obtained under the assumption of an approximately constant slope of the upper boundary surface of the layer at a distance of the order of the wavelength. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 30–41, January–February, 2009.     

Theory of stability of rotating shear flows

The stability of rotating horizontal-shear flows is investigated within the framework of the linear approximation. The shear flow perturbations are divided into three classes (symmetric and two- and three-dimensional) and sufficient conditions of stability are obtained for each class. The perturbation dynamics in a flow with constant horizontal shear are described and the algebraic instability of the flow with respect to three-dimensional perturbations is detected. It is shown that the symmetric perturbations may be localized (trapped) inside the shear layer. The problem of finding the growth rates and frequencies of the trapped waves is reduced to a quantum-mechanical Schrödinger equation. Exact solutions are obtained for a “triangular” jet and hyperbolic shear.     

Translation of a sphere in a rotating viscous fluid: A numerical study

The translation of a sphere moving along the axis of a rotating viscous fluid is studied by the finite difference method at moderate Reynolds (up to R = 500) and Taylor (up to T = 100) numbers. Suppression of the separation is observed with increasing rotation parameter T. The drag coefficient is also presented. It is observed that the drag coefficient is less than that with no rotation in the range 0<N<0·7, where N = 2T/R is the inverse Rossby number. The same phenomenon was observed experimentally by Maxworthy in the range 0<N<0·75±0·03.     

Longwave Approximation in Film Flow Theory

An asymptotic longwave model which takes dispersive terms into account is constructed for describing the motion of thin films with finite deviations from the middle surface. An exact periodic solution describing a nonlinear capillary wave is constructed within the framework of the model. Small deviations from the nonlinear capillary wave are described by a linear system with periodic coefficients. It is shown that for wave perturbation periods greater than a certain critical value the monodromy matrix of this system has eigenvalues whose absolute values are equal to unity. For perturbation periods less than the critical period the absolute value of one of the eigenvalues becomes greater than unity.     

Flow of a Casson fluid on a rotating disk

The flow of a thin layer of a Casson fluid on a fast rotating disk is considered. The film thickness distribution at various times for various initial thickness distribution is calculated. The stability of the flow is examined.     

MHD rotating flow of a viscous fluid over a shrinking surface

This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows.     

Axisymmetric magnetohydrodynamic flow of Jeffrey fluid over a rotating disk

This paper examines the magnetohydrodynamic boundary layer flow of Jeffrey fluid due to a rotating disk. The governing partial differential equations are first transformed into the coupled system of ordinary differential equations and then solved by using the homotopy analysis method. The influence of various involved physical parameters on the dimensionless radial and azimuthal velocities is sketched and analyzed. The variation of skin friction coefficients in radial and azimuthal directions is studied for various values of pertinent parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

Disturbances to a rotating fluid with a sphere moving along the axis of rotation

I.IntroductionWhenabodyprojectedinthewaterisintranslationalmotionthroughwateritwillcertainlycausesdisturbances.Howtodetectthedisturbancesf'arawayfromthebodyandhowtodetermillebytheintbrlnationobtainedthepositionandvelocityofthebodyandthesiteofprojectionareimportant.Astheprojectedbodyisrotating,thefluidaroundisinauniformlyrotationrelativetothereferenceframerotatinginsynchronismwiththebody,andthisisaquestionofthemotionofabodyinarotatingfluid.SuchproblemswerefirststudiedbyProudman(1916)l']andTay…     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.     

Nonlinear waves in a liquid film on a near-horizontal surface

Nonlinear waves in a liquid film on a slightly inclined rigid plane are studied. A mathematical model is reduced to a system of two evolutionary equations for the layer thickness and the local fluid mass flow. In addition to viscous forces, gravity, and surface tension, the pressure difference over the layer thickness, induced by the gravity force projection on the normal to the underlying surface, is also taken into account. Spatially periodic solutions developing with time from small initial disturbances into regular nonlinear waves are considered. A spectral representation of the solution, the Galerkin method with respect to the uniform coordinate, and subsequent numerical calculation of the corresponding dynamic system on large time intervals are employed. Different variants in the space of the three governing parameters are calculated and some basic mechanisms of nonlinear dynamics of the two-dimensional waves are detected. The calculation results are compared with the existing experimental data. It is shown that the theoretical conclusions can be used to interpret and predict experiments.     

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

Instability in gravity-driven flow over uneven permeable surfaces

We develop a theoretical model for inclined free-surface flow over a porous surface exhibiting periodic undulations. The effect of bottom permeability is incorporated by imposing a slip condition that accounts for the nonplanar geometry of the fluid–porous medium interface. Under the assumption of shallow flow, equations of motion accounting for inertial effects are obtained by retaining in the Navier-Stokes equations terms that are up to second-order with respect to a small shallowness parameter. The explicit dependence on the cross-stream coordinate is eliminated from these equations by means of a weighted residual procedure. A linear stability analysis of the steady flow is performed in connection with Floquet–Bloch theory. The results predict that bottom permeability has a destabilizing influence on the flow. A physical explanation has been proposed which involves examining how permeability affects the steady-state flow. Conclusions are drawn regarding the combined effect of the surface tension of the fluid and the parameters describing the bottom surface including permeability, inclination and the amplitude and wavelength of the undulations that generate the bottom topography. A numerical scheme for solving the fully nonlinear governing equations is also outlined. The instability of particular steady flows is determined by conducting nonlinear simulations of the temporal evolution of the flow and comparisons are made with the predictions from the linear analysis. Comparisons with existing experimental data are also included.     

Onset of self-oscillations upon the loss of stability of spatially periodic two-dimensional viscous fluid flows relative to long-wave perturbations

The long-wave asymptotics of the secondary flow that arises after a steady, spatially periodic flow loses stability are studied when one of the periods tends to infinity and the rate of base flow along the longer period is equal to zero. It is shown that if certain non-degeneracy conditions are satisfied, then from the base solution a self-oscillatory regime branches off and both hard and soft stability loss is possible. For the leading terms of the asymptotics explicit formulas are obtained. Examples of self-oscillations calculated for specific flows are presented and the behavior of the fluid particle trajectories in the self-oscillatory regime branching off from the base flow is investigated.     

Thin film flow of non‐Newtonian MHD fluid on a vertically moving belt

Thin film flow of an Oldroyd 6‐constant fluid on a vertical moving belt is investigated analytically and numerically. The governing equations for the flow field are derived for a steady one‐dimensional flow. The effect of constant applied magnetic field is included and its influence on the flow field is studied. The nonlinear governing equations are solved analytically and the exact solution is obtained in an elegant way. Numerical solutions are also obtained using higher‐order Chebyshev spectral methods. The influence of various non‐Newtonian parameters, gravitational force and applied magnetic field is investigated. The results showing the effect of gravity, magnetic field and material constants α₁ and α₂ are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

A computational study on robust prediction of transition point over NACA0012 aerofoil surfaces from laminar to turbulent flows

Flowtransition from laminar toturbulent is prerequisite todecide whereabouts to apply surface flowcontrol techniques. This appears missing in a number of works in which thecontrol effects were merelyinvestigated without getting insight into alteration of transition position. The aim of this study is to capture the correctposition of transition overNACA0012 aerofoil at different angles of attack. Firstly, an implicit, time marching, highresolution total variation diminishing (TVD) scheme was developed to solve the governingNavier—Stokes equations forcompressible fluid flows around aerofoil sections to obtain velocity profiles around the aerofoilsurfaces. Secondly, the linear instability solver based on the Orr—Sommerfeld equations and the e^N methods were developed to calculate the onset of transition over the aerofoil surfaces. Forthe low subsonic Mach number of 0.16, the accuracy of the compressible solutions was assessed bysome available experimental results of low speed incompressible flows. In allcases, transition positionswere accurately predicted which shows applicability and superiority of the present work to beextended for higher Mach number compressible flows. Here, transition prediction methodology is described and the results of this analysiswithout active flow controlor separation are presented.